Collective neutrino oscillations and heavy-element nucleosynthesis in supernovae:
exploring potential effects of many-body neutrino correlations

Here, we shall describe how the charged-current neutrino and antineutrino capture rates are calculated in this simplified neutrino problem. These rates depend on the neutrino number fluxes across the energy spectrum and the corresponding cross-sections. For example, for $\nu_{e}$ capture on neutrons, one has

where $\sigma_{\nu_{e}n}$ is the capture cross-section and $dn_{\nu_{e}}(\bm{r},t,\bm{p})$ is the differential number density of electron neutrinos with momentum $\bm{p}$, at location $\bm{r}$ and time $t$. The limits of integration on the neutrino momentum directions in Eq. (1) are dependent on the geometry of the neutrino source. In the rest of the discussion, we shall assume that there is no explicit time-dependence in the problem, so that the geometry and the neutrino emission characteristics are fixed with respect to time²²2From the point of view of the flavor evolution calculations, the time-independent approximation can be justified on account of the neutrino crossing timescale across the supernova envelope being much shorter than the protoneutron star cooling timescale over which the neutrino emission characteristics change significantly. From the point of view of nucleosynthesis, a justification of this approximation requires the relevant expansion timescales of the neutrino driven wind to also be shorter than the PNS cooling timescale. This happens to be the case for the outflow trajectories considered here.. Hence, the label $t$ shall be omitted henceforth. The neutrino capture rates do still have an implicit time dependence, encoded in the motion $r(t)$ of a fluid element as it travels outward from the proto-neutron star.

Here it is useful to define the quantity $dn_{\nu,\alpha}(\bm{r},\bm{p})$ as the differential number density of the subset of neutrinos at location $\bm{r}$ with momentum $\bm{p}$, which were emitted specifically with an initial flavor $\alpha$. In other words, $dn_{\nu,\alpha}(\bm{r},\bm{p})$ would represent the differential neutrino number density in flavor $\alpha$, in the absence of any flavor evolution. One can then write $dn_{\nu_{e}}$ in terms of $dn_{\nu,\alpha}$ for each flavor and the corresponding flavor conversion probabilities $P_{\alpha e}$ as follows:

In what follows, we shall assume a spherically symmetric neutrino source with a sharp decoupling surface at $r=R_{\nu}$ and isotropic emission in all outward directions from each point on the surface [i.e., the neutrino bulb geometry from Duan et al. [2006]]. Under this assumption, the differential number densities $dn_{\nu,\alpha}$ lose their dependence on the direction of the neutrino momenta and can instead be written simply in terms of the neutrino energy, as:

where $f_{\nu,\alpha}(E)$ is the initial occupation number as a function of energy, and $\vartheta$ and $\phi$ are the polar and azimuthal angles, respectively, relative to the radial direction at distance $r$ from the center of the neutrino source. With these symmetry assumptions, $d\phi$ integrates to $2\pi$, and the limits of integration for $\cos\vartheta$ depend on $r$ (e.g., see Fig. 1).

Flavor oscillations in dense neutrino streams may be substantially impacted by coherent $\nu$-$\nu$ forward scattering. This interaction causes the flavor evolution histories of neutrinos with different energies/momenta to become coupled to one another. Moreover, these interactions may also lead to quantum entanglement among neutrinos. In essence, an interacting neutrino system undergoing flavor evolution is a quantum many-body system described by a Hamiltonian with one- and two-body interactions, the latter of which are long-range in momentum space. The ‘one-body’ terms describe vacuum oscillations of each neutrino as well as neutrino coherent forward scattering with ordinary matter (electrons and nucleons) [Wolfenstein, 1978, Mikheyev & Smirnov, 1985], whereas the two-body term represents $\nu$-$\nu$ interactions [Pantaleone, 1992b, a]. In regimes where $\nu$-$\nu$ interactions dominate, the neutrino-matter forward scattering term can often be “rotated away” with a suitable change-of-basis transformation, at the expense of an in-medium modification to the mixing angle and mass-squared splitting. For simplicity, we neglect this term in our calculation. For a system of $N$ neutrinos with discrete momenta (and, therefore, discrete energies) quantized as plane waves in a box of volume $V$, this Hamiltonian in a two-flavor approximation can be written as:

where $\omega_{\bm{p}}\equiv\delta m^{2}/(2|\bm{p}|)$ are the vacuum oscillation frequencies of the neutrinos (with $\delta m^{2}$ being the mass-squared splitting). $\vec{J}_{\bm{p}}$ is a vector of operators forming an SU(2) algebra and representing the neutrino “isospin” in the flavor or mass basis [for details, see Balantekin & Pehlivan [2007], Pehlivan et al. [2011]]. Additionally, $\mu_{\bm{pq}}\equiv(\sqrt{2}G_{F}/V)\,(1-\cos{\theta_{\bm{pq}}})$ is the interaction strength between the neutrinos ³³3Some studies define the interaction strength as $\mu_{\bm{pq}}\equiv(\sqrt{2}G_{F}N/V)$, with the Hamiltonian terms then being proportional to $\mu_{\bm{pq}}/N$, thus achieving the same final outcome., which in general depends on the intersection angle $\theta_{\bm{pq}}$ between their momenta, and $\vec{B}=(0,0,-1)$ in the mass basis and $(\sin{2\theta},0,-\cos{2\theta})$ in the flavor basis, where $\theta$ is the neutrino mixing angle.

We assume the neutrinos to be produced in an initially uncorrelated state at the PNS surface, and therefore their $N$-body wave function has a direct product form:

Since the neutrinos are produced by weak interactions, one typically assumes that each $\ket{\psi_{\bm{p}}}$ is initially in a flavor state (i.e., $\ket{\nu_{e}}$ or $\ket{\nu_{x}}$ in the two-flavor approximation, where $\ket{\nu_{x}}$ is a superposition of $\ket{\nu_{\mu}}$ and $\ket{\nu_{\tau}}$). The wave function $\ket{\Psi}$ evolves in time according to the Schrödinger equation⁴⁴4Previously, we stated the assumption of steady-state neutrino emission, removing any explicit time-dependence from the problem. With that assumption, “time” $t$ in the Schrödinger equation can be treated as a proxy for location $\vec{r}$, with $t=0$ corresponding to $|\vec{r}|=R_{\nu}$. Moreover, in this spherically symmetric single-angle picture, we follow only radially directed trajectories for flavor evolution and nucleosynthesis.:

where $H$ is the many-body neutrino Hamiltonian from Eq. (4). In general, the $\nu$–$\nu$ interaction term of the Hamiltonian can drive the evolution of the wave function away from a separable product state, resulting in development of nontrivial quantum entanglement between the constituent neutrinos. In this situation, a complete description of the many-body evolution as a function of time requires a Hilbert space that is $2^{N}$-dimensional, where $N$ is the number of neutrinos.

The flavor evolution of individual neutrinos in the ensemble can be quantified in terms of the expectation values of certain one-body operators within each individual neutrino subspace—for instance, the components of the isospin $\vec{J}_{\bm{q}}$ in the flavor or mass basis. One can define the neutrino “Polarization vectors” $\vec{P}_{\bm{q}}=2\,\langle\vec{J}_{\bm{q}}\rangle$, which contain information about the flavor composition of the neutrino, with $P_{z}$ denoting the net lepton number in the flavor or mass basis and the $P_{x}$ and $P_{y}$ components representing the coherence between different flavor or mass eigenstates. The relation between the Polarization vector components in the flavor ($f$) and mass ($m$) bases is given by:

where $\theta$ is the neutrino mixing angle. The electron flavor survival probability, which is required for calculating the (anti-)neutrino capture rates, can then be obtained using $P_{\alpha e}=(1+\text{sign}(\omega)\,P_{z}^{(f)})/2$. Here, $\text{sign}(\omega)=+1$ for neutrinos and $-1$ for antineutrinos.

The exponential scaling of the complexity of this many-body system as a function of particle number makes this problem numerically intractable even for a relatively small number ($\sim$ a few tens) of neutrinos in the most general case. To circumvent this difficulty, a frequently adopted approach involves neglecting any nontrivial quantum correlations among the neutrinos, thereby enforcing that the neutrino wave function (initially in a product state) forever remains in a product state⁵⁵5Quantum Kinetic treatments [e.g., Sigl & Raffelt [1993], Vlasenko et al. [2014]] that include both coherent flavor evolution and incoherent collisional effects do implicitly capture some effects of multi-neutrino quantum correlations up to $\mathcal{O}(G_{F}^{2})$. Here, “mean-field” refers to just the coherent part of the quantum kinetic description.. Effectively, this simplification amounts to replacing operator products in the Hamiltonian with one-body terms proportional to an operator times an expectation value. One can then reduce the Hamiltonian to an effective one-body operator form [Balantekin [2018]]:

Ehrenfest’s theorem then dictates that the expectation values of the isospin operators, $\langle\vec{J}_{\bm{q}}\rangle\equiv\bra{\Psi}\vec{J}_{\bm{q}}\ket{\Psi}$ evolve according to:

Using the SU(2) commutation relations for the isospin operators, together with the mean-field condition from Eq. (10), one obtains the mean-field evolution equations for the neutrino polarization vectors, $\vec{P}_{\bm{q}}$:

Certain practical differences between the many-body and the mean-field treatments can be quantified by comparison of their predictions for the evolution of Polarization vectors. In the many-body treatment, one can use Eq. (6), together with the Hamiltonian from Eq. (4), to evolve the many-body wave function $\ket{\Psi}$ and then compute the polarization vectors $\vec{P}_{\bm{q}}=2\,\langle\vec{J}_{\bm{q}}\rangle$ at each time step. These results can then be compared to the Polarization vectors of the corresponding mean-field solution, obtained by integrating Eq. (12).

Neutrinos play a vital role in supernova neutrino-driven wind nucleosynthesis processes through capture reactions on free nucleons:

These capture reactions shape the neutron-to-proton ratio ($n_{n}/n_{p}$) and the electron fraction $Y_{e}=1/(1+n_{n}/n_{p})$ in the initial phase of the outflow (at high temperature and density), which determine what type of nucleosynthesis is possible in the ejecta, and influence the availability of free nucleons to capture as element synthesis proceeds. We start each nucleosynthesis calculation from $T\sim 10$ GK, when both electron and positron capture rates are very small compared to neutrino capture rates and the ejecta are in a weak equilibrium stage. At this stage, the electron fraction $Y_{e}$ can be approximated as

In this work, we have investigated the nucleosynthesis outcomes for different choices of initial neutrino energies (or equivalently, oscillation frequencies $\omega_{q}$), roughly based on supernova simulations (e.g., Keil et al. [2003], Mirizzi et al. [2016], Horiuchi et al. [2018]). Broadly, these choices are classified as ‘symmetric’ (sym) or ‘asymmetric’ (asym), based on whether the initial $\bar{\nu}_{e}$ average energies are equal to or greater than the initial $\nu_{e}$ average energies. For the calculations with $N=4$ neutrino modes, we assign one mode each to the $\nu_{e}$, $\bar{\nu}_{e}$, $\nu_{x}$, and $\bar{\nu}_{x}$ species, respectively ($N_{\nu,\alpha}=1$ for each $\alpha$). For the $N=8$ case, we assign two modes to each of the species ($N_{\nu,\alpha}=2$). Table 2 summarizes the parameters for each of these initial neutrino distribution choices: the energies $E_{\nu,\alpha}$ and the corresponding luminosities $L_{\nu,\alpha}$ such that Eq. (14) is satisfied, given the values of $R_{\nu}$ and $\mu(R_{\nu})$ from Table 1. The resulting initial electron fractions to be used for the heavy-element nucleosynthesis simulations are also included in the final column.

To test the potential effects of neutrino interactions on the nucleosynthesis results, for each set of calculations, we consider four different neutrino treatments: no neutrinos involved for $T<10$ GK (nn), no neutrino oscillations (nosc), mean-field neutrino oscillations (mf) described in Sec. 2.3, and many-body neutrino oscillations (mb) described in Sec. 2.2. The default neutrino number used for our calculations is 2 $\nu$ + 2 $\bar{\nu}$; we also tested a symmetric (sym) and an asymmetric (asym2) scenario with an increased neutrino number: 4 $\nu$ + 4 $\bar{\nu}$. Here we adopted the normal mass ordering for the neutrino flavor evolution as default (i.e., $\omega_{\bm{p}}>0$ for neutrinos and $<0$ for anti-neutrinos); the effect of inverted mass hierarchy scenario is also examined, as discussed in Section 5.1.2.

Figure 2 shows an example comparison between the mb and mf calculations for the flavor evolution of a 2 $\nu$ + 2 $\bar{\nu}$ neutrino system in the “sym” configuration. In particular, we show the evolution (i.e., with distance $r$) of neutrino polarization vector component $P_{z}(\omega)$ in the mass basis, evolution of the average $\nu_{e}$ and $\bar{\nu}_{e}$ energies, as well as of the $\nu_{e}$ and $\bar{\nu}_{e}$ capture rates. The vacuum oscillation frequencies of the neutrinos, calculated using their average energy through $\omega_{j}=\delta m^{2}/2E_{j}$, are $\omega_{1}$, $\omega_{2}$, $\omega_{3}$, and $\omega_{4}$ for neutrinos initially in the $\bar{\nu}_{e}$, $\bar{\nu}_{x}$, $\nu_{x}$, and $\nu_{e}$ states, respectively. We can see that the onset of high-amplitude oscillations occurs earlier for the mb case, and the two cases have distinct evolution with distance from the PNS, which we anticipate will influence the ultimate nucleosynthesis yields.

In this work we adopt the nuclear reaction network code Portable Routines for Integrated nucleoSynthesis Modeling (PRISM) [Mumpower et al., 2018, Sprouse et al., 2020]. The four neutrino treatments summarized in Sec. 4.1 (nn, nosc, mf, mb) are implemented into the nucleosynthesis calculations in the form of external $\nu_{e}+n$ and $\bar{\nu}_{e}+p$ capture rates, for each of the neutrino energy configurations from Table 2. At $T\sim 10$ GK, the nuclear composition is determined by Nuclear Statistical Equilibrium (NSE). Here we obtain the initial composition at $\sim 10$ GK through an NSE calculation with $Y_{e}$ obtained from Eq. (19) and the density taken from the chosen matter trajectory. We dynamically evolve the composition and the electron fraction from $T\sim 10$ GK onwards. This naturally incorporates a possible ‘alpha effect’—an increase in the number of alpha particles and, subsequently, seeds and corresponding decrease in free nucleons available for capture—due to neutrino interactions on free nucleons during alpha particle formation [Fuller & Meyer, 1995, Meyer et al., 1998]. For the network calculation, we use REACLIB reaction rates [Cyburt et al., 2010] along with NUBASE $\beta$-decay properties [Kondev et al., 2021]. The available REACLIB data is supplemented with neutron capture rates, $\beta$-decay rates, and fission properties in the actinide region based on the FRDM/FRLDM model [Mumpower et al., 2018], though we find the nucleosynthesis calculations in this work largely do not extend into this region of the nuclear chart.

To test the nucleosynthetic outcomes of both the symmetric and asymmetric neutrino calculations, here we adopt two types of parameterized supernova neutrino-driven wind trajectories: trajectories parameterized as per the “standard model”  in Wanajo et al. [2011] (Wanajo2011) with a range of entropies, and a high-entropy trajectory from Duan et al. [2011] (Duan2011). The details of these trajectories are summarized in Table 3. We also tested the shocked trajectories from Arcones et al. [2007a]; the resulting abundance patterns lie between those for the Wanajo2011 trajectory with entropy per nucleon in units of the Boltzmann constant $s/k=50$, and those for the Duan2011 trajectory, so for brevity these results do not appear in this work.

The Wanajo2011 trajectories adopted the wind solution for the “standard model” in Wanajo et al. [2011] with $M_{\rm ns}=1.4\,M_{\odot}$ and $L_{\nu}=1\times 10^{52}$ erg/s, followed by a subsonic expansion after wind termination at 300 km using a standard parameterization with the time evolution of temperature $T$ and density $\rho$ as follows: $\rho(t)\propto t^{-2}$, and $T(t)\propto t^{-2/3}$. We assume an adiabatic expansion throughout the evolution, with constant values for the entropy per baryon set to $s/k=50$, 100, and 150. Higher (lower) values of initial entropy result in lower (higher) mass fractions of seed nuclei produced, with more (fewer) free nucleons remaining for capture. The Duan2011 trajectory is more extreme, with faster expansion and a higher entropy of $s/k\sim 200$. The density initially falls exponentially, $\rho(t)\propto e^{-t/3\tau}$, with $\tau\sim 12$ ms for $3\,\text{GK}\leq T\leq 6\,\text{GK}$ and $\tau\sim 18$ ms for $1.5\,\text{GK}\leq T\leq 3\,\text{GK}$, before switching to a power law $\rho(t)\propto t^{-2}$ when the temperature is roughly 1 GK. This $\sim t^{-2}$ drop-off, rather than the much faster $\rho\propto t^{-3}$ that one expects in the event of free expansion, can be attributed to the fact that the neutrino-driven wind is expected to encounter the outer layers of the supernova ejecta/envelope as it emerges from the inner “hot-bubble” region, slowing down its expansion. The entropy of this trajectory is somewhat outside of those expected from modern supernova simulations. However, this combination of fast expansion and high entropy maximizes the ratio of free nucleons to seeds, which is ideal for emphasizing the nucleosynthetic impact of the differences in neutrino treatments.

Figure 3 illustrates the evolution of the neutrino capture rates as a function of the temperature along the matter trajectory for the Duan2011 and Wanajo2011 $s/k=50$ cases, respectively. The neutrino fluxes are taken from the symmetric (sym) case with many-body (mb) and mean-field (mf) oscillation treatments. In all cases, the onset of oscillations results in a steep rise to the capture rates, which occurs within the temperature window relevant for nucleosynthesis. The degree and duration of the rise are expected to shape the formation of seed nuclei and the subsequent nucleosynthesis. Note that, in each of our calculations, we assume steady-state neutrino emission from the source, resulting in an over-estimation of about 20% in the integrated $\bar{\nu}_{e}$ capture rates for the Wanajo trajectories, and about 2% for the Duan2011 trajectory, when compared with corresponding calculations that used exponentially decaying neutrino luminosities with a time constant of 3 s. In each case, this approximation does not qualitatively affect any of the results and conclusions.

Symmetric neutrino energies and luminosities between the $\nu_{e}$ and $\bar{\nu}_{e}$ species will produce charged-current neutrino capture rates that are faster than antineutrino rates, thanks to the neutron-proton mass difference. As a result, the resulting nucleosynthetic conditions in the supernova neutrino-driven wind will tend to be proton-rich. If a sufficient excess of free protons is available following seed formation, along with a sizeable $\bar{\nu}_{e}$ flux to convert a fraction of the free protons into neutrons for $(n,p)$ and $(n,\gamma)$ reactions, then a $\nu p$ process will result.

The asymmetric neutrino calculations adopted in Table 2 result in capture rates for antineutrinos that are similar to or higher than for neutrinos, producing equilibrium electron fractions that are closer to or less than 0.5. As already noted, the supernova neutrino-driven wind could be a mildly neutron-rich environment that allows for a limited or weak $r$ process, with the nucleosynthesis extent sensitive to the neutrino physics adopted. So, here we investigate a possible $r$ process in the neutrino-driven wind and the effect of the neutrino physics treatments described above on the $r$-process yields. We examine the nucleosynthetic impact of neutrinos with asymmetric energy distributions and luminosities with four different choices of neutrino energy distributions, i.e., asym2, asym2.1, asym3, and asym4, corresponding to equilibrium $Y_{e}$ values ranging from 0.504 to 0.366, as listed in Table 2.

Three of the four adopted sets of asymmetric neutrino energies result in initially neutron-rich conditions. During alpha particle formation, all of the protons are consumed to produce alpha particles. This reaction leaves few free protons to convert to neutrons via antineutrino capture, while any excess free neutrons are still subject to neutrino captures. The protons produced as a result combine promptly with free neutrons to continue forming alphas. Thus neutrino interactions act to increase the number of alpha particles (and, subsequently, seed nuclei) and reduce the number of free neutrons available for capture on the seed nuclei, which is the so-called alpha effect [Fuller & Meyer, 1995, McLaughlin et al., 1996, Meyer et al., 1998]. With the influence of collective neutrino oscillations included, the effective energies of $\nu_{e}$ and $\bar{\nu}_{e}$ are raised. The corresponding increases to the neutrino capture rates cause the alpha effect to become more significant and the production of heavy nuclei is hindered.

We first test the nucleosynthesis calculations with the asymmetric neutrino treatment asym4, which results in the most neutron-rich conditions of the four cases and therefore is most likely to produce $r$-process species. The corresponding abundance patterns and the nucleosynthesis paths are shown in Fig. 14, for the $s/k=50$, 100, and 150 Wanajo11 matter trajectories. All simulations result in at most a weak $r$ process, and only the $s/k=150$ case robustly produces the second ($A\sim 130$) $r$-process peak. The influence of neutrinos and their oscillations only act to hinder the $r$ process through an enhanced alpha effect. The asym4-mb case shows the largest influence, though overall the impact of neutrinos on the ultimate nucleosynthetic outcome is much smaller compared to the simulations discussed in Sec. 5.1. The free nucleon-to-seed ratio is low and the neutrons are efficiently consumed, cutting off the potential influence of neutrinos. This situation is in stark contrast to the proton-rich examples above in which protons are available to convert to neutrons throughout the element synthesis.

In order to fully explore the potential impact of the different neutrino treatments (nn, nosc, mf, mb) defined in Sec. 4.1, we also calculate the nucleosynthesis resulting from the Duan2011 matter trajectory. Its fast expansion and high entropy ensure that a robust main $r$ process—one that reaches $A\sim 195$ peak nuclei—is possible for at least some of the neutrino energy distributions explored, and the impact of the neutrino treatments is maximized for all asymmetric configurations explored: asym4, asym3, asym2.1, and asym2.

For the most neutron-rich asym4 case, as seen in the right panel of Fig. 15, the nucleosynthesis calculation for the Duan2011 matter trajectory with neutrino interactions turned off for $T<10$ GK results in a robust main $r$-process pattern that reaches the actinide region. Neutrino interactions produce an alpha effect of varying strength, as shown in Fig. 16, with the asym4-mb case resulting in the lowest free neutron abundance and highest alpha abundance. As a result, we observe decreased production of heavier nuclei especially around the third-peak region and beyond, with the smallest $A\sim 195$ peak produced in the asym4-mb case.

The asym3 case starts with a slightly higher $Y_{e}$ value at equilibrium, producing a lower initial neutron abundance and a faster consumption of neutrons than in the asym4 case, as shown in Figure 16. As a result, the abundance pattern for the asym3-nn case just barely fills out the third-peak region and underproduces the actinides. Again, from the left panel of Fig. 15, we can see that neutrinos and neutrino oscillations hinder the $r$-process production of heavier nuclei. The effect of the difference in neutrino treatments on the $r$-process yields follows the same pattern as the asym4 case, but with more significant deviations shown.

Finally, we examine the effects of the different neutrino treatments on the nucleosynthesis calculations for the Duan2011 matter trajectory with initial $Y_{e}$ values around 0.5, with the results shown in Fig. 17. For the neutrino parameters that result in an electron fraction slightly bigger than 0.5, the nucleosynthesis result is either iron-peak nuclei (for asym2-nn), or a mild $\nu p$ process when neutrino interactions are included. Similar to the symmetric neutrino cases described in Section 5.1.1 above, the extent of the $\nu p$ process is sensitive to the treatment of neutrino oscillations, with the asym2-mb case producing the highest $\nu p$-process yields compared to the other neutrino treatments. The asym2.1 neutrino distribution on the other hand corresponds to an equilibrium electron fraction slightly smaller than 0.5 and result in a weak $r$ process up to the second $r$-process peak. Here the influence of neutrinos is modest, similar to the Wanajo2011 asym4 results shown in Fig. 14.

For the asymmetric neutrino conditions studied here, we find the neutrino physics to be particularly crucial for the nucleosynthesis outcomes that sit at a threshold, either for an $r$ process that barely reaches the third peak region, or for initial conditions around $Y_{e}=0.5$. We accordingly expect similar behavior for an $r$ process that proceeds just beyond the second peak. In these cases, the changes to the neutrino interaction rates that result from the treatment of neutrino oscillation behavior can produce marked changes in the abundance yields. However, in general, the leverage that the neutrinos have on nucleosynthetic outcomes in neutron-rich conditions remains smaller than in very proton-rich conditions. This difference is due to neutrino interactions changing the abundance of the sub-dominant free nucleon species, e.g., neutrons in proton-rich and protons in neutron-rich nucleosynthesis, as element synthesis proceeds. However, as the temperature drops, only neutrons can be captured (and protons cannot). Thus, the influence of neutrino interactions in proton-rich conditions is extended throughout the element synthesis, resulting in a larger impact on a $\nu p$ process compared to the $r$ process for the collective oscillation effects considered here.